Projective variety
In algebraic geometry, a projective variety is an algebraic variety that is a closed subvariety of a projective space. That is, it is the zero-locus in of some finite family of homogeneous polynomials that generate a prime ideal, the defining ideal of the variety.
A projective variety is a projective curve if its dimension is one; it is a projective surface if its dimension is two; it is a projective hypersurface if its dimension is one less than the dimension of the containing projective space; in this case it is the set of zeros of a single homogeneous polynomial.
If X is a projective variety defined by a homogeneous prime ideal I, then the quotient ring
is called the homogeneous coordinate ring of X. Basic invariants of X such as the degree and the dimension can be read off the Hilbert polynomial of this graded ring.
Projective varieties arise in many ways. They are complete, which roughly can be expressed by saying that there are no points "missing". The converse is not true in general, but Chow's lemma describes the close relation of these two notions. Showing that a variety is projective is done by studying line bundles or divisors on X.
A salient feature of projective varieties are the finiteness constraints on sheaf cohomology. For smooth projective varieties, Serre duality can be viewed as an analog of Poincaré duality. It also leads to the Riemann–Roch theorem for projective curves, i.e., projective varieties of dimension 1. The theory of projective curves is particularly rich, including a classification by the genus of the curve. The classification program for higher-dimensional projective varieties naturally leads to the construction of moduli of projective varieties. Hilbert schemes parametrize closed subschemes of with prescribed Hilbert polynomial. Hilbert schemes, of which Grassmannians are special cases, are also projective schemes in their own right. Geometric invariant theory offers another approach. The classical approaches include the Teichmüller space and Chow varieties.
A particularly rich theory, reaching back to the classics, is available for complex projective varieties, i.e., when the polynomials defining X have complex coefficients. Broadly, the GAGA principle says that the geometry of projective complex analytic spaces is equivalent to the geometry of projective complex varieties. For example, the theory of holomorphic vector bundles on X coincide with that of algebraic vector bundles. Chow's theorem says that a subset of projective space is the zero-locus of a family of holomorphic functions if and only if it is the zero-locus of homogeneous polynomials. The combination of analytic and algebraic methods for complex projective varieties lead to areas such as Hodge theory.
Variety and scheme structure
Variety structure
Let k be an algebraically closed field. The basis of the definition of projective varieties is projective space, which can be defined in different, but equivalent ways:- as the set of all lines through the origin in
- as the set of tuples, with not all zero, modulo the equivalence relation for any. The equivalence class of such a tuple is denoted by This equivalence class is the general point of projective space. The numbers are referred to as the homogeneous coordinates of the point.
does not make sense for arbitrary polynomials, but only if f is homogeneous, i.e., the degrees of all the monomials are the same. In this case, the vanishing of
is independent of the choice of.
Therefore, projective varieties arise from homogeneous prime ideals I of, and setting
Moreover, the projective variety X is an algebraic variety, meaning that it is covered by open affine subvarieties and satisfies the separation axiom. Thus, the local study of X reduces to that of an affine variety. The explicit structure is as follows. The projective space is covered by the standard open affine charts
which themselves are affine n-spaces with the coordinate ring
Say i = 0 for the notational simplicity and drop the superscript. Then is a closed subvariety of defined by the ideal of generated by
for all f in I. Thus, X is an algebraic variety covered by open affine charts.
Note that X is the closure of the affine variety in. Conversely, starting from some closed variety, the closure of V in is the projective variety called the of V. If defines V, then the defining ideal of this closure is the homogeneous ideal of generated by
for all f in I.
For example, if V is an affine curve given by, say, in the affine plane, then its projective completion in the projective plane is given by
Projective schemes
For various applications, it is necessary to consider more general algebro-geometric objects than projective varieties, namely projective schemes. The first step towards projective schemes is to endow projective space with a scheme structure, in a way refining the above description of projective space as an algebraic variety, i.e., is a scheme which it is a union of copies of the affine n-space kn. More generally, projective space over a ring A is the union of the affine schemesin such a way the variables match up as expected. The set of closed points of, for algebraically closed fields k, is then the projective space in the usual sense.
An equivalent but streamlined construction is given by the Proj construction, which is an analog of the spectrum of a ring, denoted "Spec", which defines an affine scheme. For example, if A is a ring, then
If R is a quotient of by a homogeneous ideal I, then the canonical surjection induces the closed immersion
Compared to projective varieties, the condition that the ideal I be a prime ideal was dropped. This leads to a much more flexible notion: on the one hand the topological space may have multiple irreducible components. Moreover, there may be nilpotent functions on X.
Closed subschemes of correspond bijectively to the homogeneous ideals I of that are saturated; i.e., This fact may be considered as a refined version of projective Nullstellensatz.
We can give a coordinate-free analog of the above. Namely, given a finite-dimensional vector space V over k, we let
where is the symmetric algebra of. It is the projectivization of V; i.e., it parametrizes lines in V. There is a canonical surjective map, which is defined using the chart described above. One important use of the construction is this. A divisor D on a projective variety X corresponds to a line bundle L. One then set
it is called the complete linear system of D.
Projective space over any scheme S can be defined as a fiber product of schemes
If is the twisting sheaf of Serre on, we let denote the pullback of to ; that is, for the canonical map
A scheme X → S is called projective over S if it factors as a closed immersion
followed by the projection to S.
A line bundle on a scheme X over S is said to be very ample relative to S if there is an immersion
for some n so that pullbacks to. Then a S-scheme X is projective if and only if it is proper and there exists a very ample sheaf on X relative to S. Indeed, if X is proper, then an immersion corresponding to the very ample line bundle is necessarily closed. Conversely, if X is projective, then the pullback of under the closed immersion of X into a projective space is very ample. That "projective" implies "proper" is deeper: the main theorem of elimination theory.
Relation to complete varieties
By definition, a variety is complete, if it is proper over k. The valuative criterion of properness expresses the intuition that in a proper variety, there are no points "missing".There is a close relation between complete and projective varieties: on the one hand, projective space and therefore any projective variety is complete. The converse is not true in general. However:
- A smooth curve C is projective if and only if it is complete. This is proved by identifying C with the set of discrete valuation rings of the function field k over k. This set has a natural Zariski topology called the Zariski–Riemann space.
- Chow's lemma states that for any complete variety X, there is a projective variety Z and a birational morphism Z → X.
for any projective variety X over k. This fact is an algebraic analogue of Liouville's theorem. In fact, the similarity between complex analytic geometry and algebraic geometry on complex projective varieties goes much further than this, as is explained below.
Quasi-projective varieties are, by definition, those which are open subvarieties of projective varieties. This class of varieties includes affine varieties. Affine varieties are almost never complete. In fact, a projective subvariety of an affine variety must have dimension zero. This is because only the constants are globally regular functions on a projective variety.
Examples and basic invariants
By definition, any homogeneous ideal in a polynomial ring yields a projective scheme. In this sense, examples of projective varieties abound. The following list mentions various classes of projective varieties which are noteworthy since they have been studied particularly intensely. The important class of complex projective varieties, i.e., the case, is discussed further below.The product of two projective spaces is projective. In fact, there is the explicit immersion
As a consequence, the product of projective varieties over k is again projective. The Plücker embedding exhibits a Grassmannian as a projective variety. Flag varieties such as the quotient of the general linear group modulo the subgroup of upper triangular matrices, are also projective, which is an important fact in the theory of algebraic groups.